Calibrated Plane--Convex Microcavity for Room-Temperature Polaritons with Geometric \(g\)-Scaling

We present a plane--convex open microcavity that supports room-temperature polariton spectroscopy and offers a simple geometric handle on the coupling rate. The effective length ($L_{\mathrm{eff}}$) is absolutely calibrated from the free-spectral range, and piezo tuning is performed at near-normal incidence ($k_{\parallel}!\approx!0$) to avoid angle-induced degradation. Using spin-coated PEA$2$PbI$4$ quasi-2D perovskites, we observe clear anti-crossings in reflection, with vacuum Rabi splittings up to $71~\mathrm{meV}$ (reflection) and $87~\mathrm{meV}$ (PL) near $L{\mathrm{eff}}!\approx!3~\mu\mathrm{m}$. A linewidth-corrected analysis converts the apparent splitting into the coherent exciton--photon coupling rate $g$, revealing a robust geometric scaling $g \propto L{\mathrm{eff}}^{-1/2}$ across multiple longitudinal orders and spatial sites, consistent with the filled-mode thin-film limit where the transverse area cancels in the mode volume. The platform establishes a compact, broadly compatible testbed for room-temperature polaritons and provides a practical design rule: shortening $L_{\mathrm{eff}}$ is a reliable geometric lever to strengthen collective coupling in plane--convex microcavities.